Suppose you are given a single unit square, and you would like to completely cover the surface
of a cube by cutting up the square and pasting it onto the cube's surface.

> <b>Q1</b>. What is the largest cube that can be covered by a $1 \times 1$ square when cut into at most $k$ pieces?

The case $k=1$ has been studied, probably earlier than this reference:
 "Problem 10716: A cubical gift," *American Mathematical Monthly*, 108(1):81-82, January 2001,
solution by Catalano-Johnson, Loeb, Beebee.
<br />&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
<img src="https://i.sstatic.net/elzwA.jpg" alt="Square Wrapping Cube" /><br />
(This was discussed in an
<a href="https://math.stackexchange.com/questions/76660/">MSE Question</a>.)
The depicted solution results in a cube edge length of
$1/(2\sqrt{2}) \approx 0.35$.

As $k \to \infty$, there should be no wasted overlaps in the covering of the 6 faces,
and so the largest cube covered will have edge length
$1/\sqrt{6} \approx 0.41$.  What partition of the square leads to this optimal cover?

> <b>Q2</b>. For which value of $k$ is this optimal reached?

I have not found literature on this problem for $k>1$, but it seems likely it has been explored.
Thanks for any pointers!